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The renormalization techniques for determining the long-time asymp-totics of nonlinear parabolic equations pioneered by Bricmont, Kupiainen and Lin are shown to be eeective in analyzing nonlinear wave equations featuring both dissi-pation and dispersion. These methods allow us to recover recent results of Dix in a way which is both transparent and has(More)
Diabetes is a disease of the glucose regulatory system that is associated with increased morbidity and early mortality. The primary variables of this system are beta-cell mass, plasma insulin concentrations, and plasma glucose concentrations. Existing mathematical models of glucose regulation incorporate only glucose and/or insulin dynamics. Here we develop(More)
We use renormalization group (RG) techniques to prove the nonlinear asymptotic stability for the semi-strong regime of two-pulse interactions in a regularized Gierer-Meinhardt system. In the semi-strong limit the localized activator pulses interact strongly through the slowly varying inhibitor. The interaction is not tail-tail as in the weak interaction(More)
Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New(More)
The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the(More)
We use a multi-scale analysis to derive a sharp interface limit for the dynamics of bilayer structures of the functionalized Cahn–Hilliard equation. In contrast to analysis based on single-layer interfaces, we show that the Stefan and Mullins–Sekerka problems derived for the evolution of single-layer interfaces for the Cahn–Hilliard equation are trivial in(More)
We present a microfield approach for studying the dependence of the orientational polarization of the water in aqueous electrolyte solutions upon the salt concentration and temperature. The model takes into account the orientation of the solvent dipoles due to the electric field created by ions, and the effect of thermal fluctuations. The model predicts a(More)
We employ global quasi-steady manifolds to rigorously reduce innnite dimensional dynamical systems to nite dimensional ows. The manifolds we construct are not invariant, but through a renormalization group method we capture the long-time evolution of the full system as a ow on the manifold up to a small residual. For the parametric nonlinear Schrr odinger(More)
We demonstrate the existence of localized oscillatory breathers for quasi-one-dimensional Bose-Einstein condensates confined in periodic potentials. The breathing behavior corresponds to position oscillations of individual condensates about the minima of the potential lattice. We deduce the structural stability of the localized oscillations from the(More)